## BRST cohomology and phase space reduction in deformation quantization.(English)Zbl 0961.53046

The BRST cohomology in the framework of deformation quantization is defined. The starting point is a star-product $$*$$ for a Poisson structure on a manifold $$M$$, defined on the algebra $$C^\infty(M)[[\lambda]]$$, and a Hamiltonian action of a Lie group $$G$$ in $$M$$. With this action a quantum momentum map $${\mathbf J}=\sum_{r=0}^\infty\lambda^r{\mathbf J}_r:M\rightarrow {\mathfrak g}^*[[\lambda]]$$ is associated, for which $${\mathbf J}_0=J$$ is a classical equivariant momentum map. The star product is taken to be quantum covariant, i.e., such that $$\langle{\mathbf J},\xi\rangle *\langle{\mathbf J},\eta\rangle - \langle{\mathbf J},\eta\rangle *\langle{\mathbf J},\xi\rangle= \text{i}\lambda \langle{\mathbf J},[\xi,\eta]\rangle$$ for all $$\xi,\eta$$ from the Lie algebra $${\mathfrak g}$$ of $$G$$. The underlying vector space for the quantum BRST algebra is the $${\mathbb{C}}[[\lambda]]$$-module $${\mathcal A}[[\lambda]]$$ of formal power series with values in the classical super Poisson algebra $${\mathcal A}=\bigwedge{\mathfrak g}^*\otimes\bigwedge{\mathfrak g}\otimes C^\infty(M)$$ which inherits the ghost number grading as well as the $${\mathbb{Z}}_2$$-grading in even and odd elements. A one-parameter family of formal associative deformations $$\star_\kappa$$ of the classical extended super Poisson algebra is defined on $${\mathcal A}[[\lambda]]$$, for which the corresponding BRST charge $$\Theta_\kappa$$ has square zero. The deformations are equivalent by explicit equivalence transformations $$S_\kappa$$. Among the family of quantum BRST operators $${\mathcal D}_\kappa$$ in $$({\mathcal A}[[\lambda]], \star_\kappa)$$, i.e., the graded star product commutators with $$\Theta_\kappa$$, the most natural from the point of Clifford algebras is the Weyl-ordered BRST operator $${\mathcal D}_W=\frac{1}{\text{i} \lambda}\text{ad}_W(\Theta_W)$$ corresponding to $$\kappa=1/2$$. Here $$\Theta_W=\Omega+{\mathbf J}$$, where $$\Omega$$ represents the Lie bracket in $${\mathfrak g}$$ and $$\text{ad}_W$$ is the supercommutator for $$\star_{1/2}$$. Under the equivalence transformations $$S_\kappa$$, the BRST charge and the BRST operator transform according to $${\mathcal D}_\kappa=S^{-1}_{\kappa-\frac{1}{2}}\circ{\mathcal D}_W\circ S_{\kappa-\frac{1}{2}}$$, $$\Theta_\kappa=S^{-1}_{\kappa-\frac{1}{2}}(\Theta_W)$$. It turns out that $${\mathcal D}_W$$ is not very encouraging concerning cohomology computations, but luckily the quantum BRST operator $${\mathcal D}_0$$, associated with the standard ordering, defines a double complex thus giving rise to a deformed version of the classical Koszul boundary operator and the classical Chevalley-Eilenberg differential. For every regular level-zero surface $$C$$ of $$J$$ one can compute the quantum BRST cohomology of $$({\mathcal A}[[\lambda]],{\mathcal D}_0)$$ by means of a deformed augmentation, i.e., a quantized version of restricting functions on $$M$$ to $$C$$. The result is that the quantum BRST cohomology is isomorphic to the Chevalley-Eilenberg cohomology of the Lie algebra $${\mathfrak g}$$ with a $${\mathfrak g}$$-module $$C^\infty(C)[[\lambda]]$$. Moreover, the quantum BRST cohomologies of the operators $${\mathcal D}_\kappa$$ are all isomorphic. Finally, a deformed version $${\mathbf{\mathcal I}}_C$$ of the vanishing ideal of the constrained surface is defined, so that its idealiser $${\mathbf{\mathcal B}}_C$$ in $$(C^\infty(M)[[\lambda]],*)$$ modulo $${\mathbf{\mathcal I}}_C$$ turns out to be naturally isomorphic to the quantum BRST algebra. Simple examples and counterexamples are also discussed.

### MSC:

 53D55 Deformation quantization, star products 81T70 Quantization in field theory; cohomological methods 53D20 Momentum maps; symplectic reduction 17B56 Cohomology of Lie (super)algebras
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